Mathematical Models of the Simplest Fuzzy Two-Term (PI/PD) Controllers Using Algebraic Product Inference

. This paper reveals mathematical models of the simplest Mamdani PI/PD controllers which employ two fuzzy sets (N: negative and P: positive) on the universe of discourse (UoD) of each of two input variables (displacement and velocity) and three fuzzy sets (N: negative, Z: zero, and P: positive) on the UoD of output variable (control output in the case of PD, and incremental control output in the case of PI). The basic constituents of these models are algebraic product/minimum AND, bounded sum/algebraic sum/maximum OR, algebraic product inference, three linear fuzzy control rules, and Center of Sums (CoS) defuzziﬁcation. Properties of all these models are investigated. It is shown that all these controllers are di ﬀ erent nonlinear PI/PD controllers with their proportional and derivative gains changing with the inputs. The proposed models are signiﬁcant and useful to control community as they are completely new and qualitatively di ﬀ erent from those reported in the literature.


Introduction
Most of the industrial processes are still controlled largely by PID controllers due to their simplicity and low cost.Though the mathematical model (transfer function) of the PID controller is well known to the control community, mathematical modeling of fuzzy PI/PD controllers was not well established until 1993.Availability of mathematical models of fuzzy controllers and their precise understanding are fundamentally important for systematic analysis and synthesis of fuzzy control systems.
Looking at the historical development on fuzzy controller modeling, one would notice from the literature that it was Ying [1] who came up with four different models for the simplest fuzzy PI controller.He revealed that "bounded difference" inference was inappropriate for control.Later, Patel and Mohan [2] and Mohan and Sinha [3] presented three more models for the same using algebraic product/minimum AND and bounded sum/maximum OR.Subsequently, Haj-Ali and Ying [4] developed a mathematical model of fuzzy PI/PD controller using nonlinear input membership functions.Moreover, Mohan and Sinha [5] presented three more models via algebraic sum OR.Very recently, Mohan [6] has made an attempt to dispel some shortcomings that did take place inadvertently in some of the publications which are actually leading to misunderstandings and confusion.
It is very important to note that the output membership functions, shown in Figure 1(a), were considered for obtaining mathematical models in all the aforementioned publications.Though these works help us find the relationship between the models of the so-called the simplest fuzzy controllers and the controllers with multifuzzy sets [7], they cannot truly represent the models of the simplest fuzzy controllers.The output membership functions should be preferably considered as shown in Figure 1(b) to obtain mathematical models of the simplest fuzzy controllers.As a result of this, new mathematical models can be obtained.
The primary objective of this paper is to continue to reveal mathematical models of the simplest fuzzy PI/PD controllers.The results presented in this paper are significant and useful to control community as some of the models developed herein are qualitatively different from the models already reported in the literature.The paper is organized as follows: the next section describes the principal components of fuzzy two-term controllers.Section 3 presents mathematical models of the simplest fuzzy controllers whose properties are included in Section 4. Finally, Section 5 concludes the paper.

Principal Components of Fuzzy Two-Term Controllers
The incremental control effort generated by a discrete-time PI controller is given by while the control effort produced by a discrete-time PD controller is described by where d(k) = e(k), the error signal The inputs are fuzzified by L-function type and Γfunction type membership functions, shown in Figure 3, whose mathematical description is given by where always.The membership functions for the normalized output are shown in Figure 1(b).Let Then, the trapezoid shrinks to a triangle when ρ = a = 0.

Control Rule Base.
The following linear control rules are considered for fuzzy PI control.

Rule base
Defuzzify The justification of the above rule base can be seen in Mohan and Sinha [5].As a matter of fact, the above rule base can also be written in the following manner.
The rule base in the above form was considered in the current literature [1][2][3][4][5].Δu N is replaced by u N for fuzzy PD control in the above rule base.We consider "algebraic product" or "minimum" AND operation, and they are defined as where i ∈ {D − , D + } and j ∈ {V − , V + }.The OR operation in rule 2 may be performed by employing one of the following.Bounded sum: Algebraic sum: Maximum:

Inference Engine.
The basic function of inference engine is to compute overall control output value based on the individual contribution of each rule in the rule base.For each rule, a degree of match is found by using any AND operator (and also any OR operator in the case of rule 2).
Based on this degree of match, the control output value in the rule consequent is modified by using "algebraic product" inference.The reference membership functions and the inferred membership functions (shown with hatching) after employing algebraic product inference are shown in Figure 4.All possible combinations of normalized inputs d N and v N are shown in Figure 5.The control rules R 1 -R 3 are used to evaluate appropriate control law in each region.

Modelling and Simulation in Engineering
The outcomes of fuzzy control rules with algebraic product/ minimum AND and bounded sum/algebraic sum/maximum OR are given for all the regions in Table 1.

Defuzzification Module. It converts the set of all inferred
control output values into a pointwise value.Defuzzification is done here using the well-known CoS method.According to this method, the crisp value of normalized control output is given by where A(•) and C(•) are, respectively, the area and centroid corresponding to the inferred output membership function.
It is easy to compute the following from Figure 4: (15)

Mathematical Models of the Simplest Fuzzy PI/PD Controllers
Here, we present mathematical models of the simplest fuzzy controllers which are derived using algebraic product/minimum AND, bounded sum/algebraic sum/maximum OR.These controllers are classified into six different classes depending upon the AND and OR operators used.

Mathematical Models in the Regions 1-4
Class 1. Algebraic product AND and algebraic sum OR: where Class 2. Algebraic product AND and bounded sum OR: where Class 3. Algebraic product AND and maximum OR: where with s 1 and s 2 as defined in Table 2(a).
Class 4. Minimum AND and algebraic sum OR: where with s 1 , s 2 , and s 3 as defined in Table 2(b).
Class 5. Minimum AND and bounded sum OR: where with s 1 , s 2 , and s 3 as defined in Table 2(b).
Class 6. Minimum AND and maximum OR: where with s 1 , s 2 , and x N (k) as defined in Table 2(c).Region
and K d D are, respectively, the proportional, integral, and derivative constants of discrete-time PI and PD controllers, and T is the sampling period.The block-diagram of fuzzy two-term controller is shown in Figure 2 in which N d , N v , N Δu , and N u represent normalization factors of the fuzzy controller, and d N (k),v N (k), Δu N (k), and u N (k) represent normalized versions of d(k), v(k), Δu(k), and u(k), respectively.The use of normalized UoD requires a scale transformation which maps the physical values of the process input variables (in the present case d(k) and v(k)) into a normalized domain.Moreover, denormalization maps the normalized value of the process output variable (here Δu N (k) or u N (k)) into its physical domain.The normalized factors which describe input normalization are denoted by N d and N v .Similarly, the denormalization factor describing output denormalization is denoted by N −1 Δu or N −1 u .The principal components of fuzzy two-term controllers are fuzzification and defuzzification modules, control rule base, and inference engine which are described in the following.2.1.Fuzzification Module.This module converts crisp values of process variables (d N , v N , Δu N or u N ) into fuzzy sets in order to make them compatible with the fuzzy set representation of the process variables in the rules.Let d * and v * be the crisp values of inputs d N and v N , respectively.Then, their fuzzified versions are the degrees of membership μ D (d * ) and μ V (v * ), where D and V are the linguistic values taken by d N and v N , respectively.

Figure 2 :Figure 3 :
Figure 2: Block diagram of a typical fuzzy PI/PD controller.

Table 2 :
(a) Signs of s 1 and s 2 .(b) Signs of s 1 , s 2 and s 3 .(c) Attributes of s 1 , s 2 , and x N (k).

Table 1 :
(a) Outcomes of antecedent part of control rules: using ap AND.(b) Outcomes of antecedent part of control rules: using mn AND.(c) Outcomes of antecedent part of control rules: using ap/mn AND. (a)

Table 3 :
Attributes of d N (k) and v N (k).Mathematical Models in the Regions 5-8.As can be seen from Table1(c), μ 0 is independent of OR operator used.Therefore, the models are also independent of OR operation.We obtain two distinct models, one for algebraic product AND (Class 1/Class 2/Class 3) and the other for minimum AND (Class 4/Class 5/Class 6), by substituting appropriate values for d N (k) and v N (k) in the expressions of Class 3 and Class 6 models as shown in Table3.3.3.Mathematical Models in theRegions 9-12.In this case, all six classes of controllers have the same unique models as stated below.